It’s always eye-opening to see the men and women behind famous works of art. And yet, while Picasso, Dali, Matisse and perhaps a handful of other “art superstars” are recognized around the world, countless other painters and sculptors of note — including some of the finest, most influential artists of the past century — are, in a sense, invisible. Their works endure; but their look — their faces, the way they held themselves — is often an utter blank in the mind of the casual art enthusiast.

Here, LIFE.com presents a series of portraits of the artists who re-imagined and redefined the way we see the world. Pollock is here, and Picasso, along with Georgia O’Keeffe and even Renoir. But can art aficionados actually identify a photograph of, say, Claes Oldenburg? Jasper Johns? Barbara Hepworth? Giorgio de Chirico?

What’s wonderful about the pictures here, including those (like the deeply shadowed silhouettes of Hopper and O’Keeffe) in which we perceive a powerful sense of the full figure, rather than discrete details, is that each photo reveals something uniquely essential about the artist. One would be hard-pressed to find two more dissimilar portraits than those of Arshile Gorky and Robert Rauschenberg; but each of those captures an aspect of the artist’s personality and sensibility (Gorky’s moody intensity; Rauschenberg’s exuberance) that, in a heartbeat, helps us see something — even if it’s only a glimpse — of the human being behind the art.